First-Order Logic (FOL or FOPC) Syntax

• User defines these primitives:
  • Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3

  • Function symbols (mapping individuals to individuals) E.g., father-of(Mary) = John, color-of(Sky) = Blue

  • Predicate symbols (mapping from individuals to truth values) E.g., greater(5,3), green(Grass), color(Grass, Green)

• FOL supplies these primitives:
  • Variable symbols. E.g., x, y

  • Connectives. Same as in PL: not (~), and (^), or (v), implies (=>), if and only if (<=>)

  • Quantifiers: Universal (A) and Existential (E)
    • Universal quantification corresponds to conjunction ("and") in that (Ax)P(x) means that P holds for all values of x in the domain associated with that variable. E.g., (Ax) dolphin(x) => mammal(x)

    • Existential quantification corresponds to disjunction ("or") in that (Ex)P(x) means that P holds for some value of x in the domain associated with that variable. E.g., (Ex) mammal(x) ^ lays-eggs(x)

    • Universal quantifiers usually used with "implies" to form "rules." E.g., (Ax) cs627-student(x) => smart(x) means "All cs627 students are smart." You rarely use universal quantification to make blanket statements about every individual in the world: (Ax)cs627-student(x) ^ smart(x) meaning that everyone in the world is a cs627 student and is smart.

    • Existential quantifiers usually used with "and" to specify a list of properties about an individual. E.g., (Ex) cs627-student(x) ^ smart(x) means "there is a cs627 student who is smart." A common mistake is to represent this English sentence as the FOL sentence: (Ex) cs627-student(x) => smart(x) But consider what happens when there is a person who is NOT a cs627-student.

    • Switching the order of universal quantifiers does not change the meaning: (Ax)(Ay)P(x,y) is logically equivalent to (Ay)(Ax)P(x,y). Similarly, you can switch the order of existential quantifiers.

    • Switching the order of universals and existentials does change meaning:
      • Everyone likes someone: (Ax)(Ey)likes(x,y)

      • Someone is liked by everyone: (Ey)(Ax)likes(x,y)